Families of lattice planes

The quantitative assessment of the degree of crystallite orientation by x-ray examination is not free of ambiguity. From a comparative analysis [23] in which results obtained from the consideration of (105) and from three different variations of equatorial reflection were compared, the conclusion was that the first procedure can lead to underrated results, i.e., to the underestimation of the orientation. However, it can be assumed that this does not result from an incorrect procedure, but from ignoring the fact that the adjacent (105) reflex can overlap. The absence of the plate effect of the orientation is characteristic of the orientation of crystallites in PET fibers. The evidence of this absence is the nearly identical azimuthal intensity distributions of the diffracted radiation in the reflexes originating from different families of lattice planes. The lack of the plate effect of orientation in the case of PET fiber stretching has to do with the rod mechanism of the crystallite orientation. 

Figure 7 (a) Bragg reflection from a set of lattice planes, (b) The cone of diffracted X-rays from a powder specimen. The cone contains all X-rays reflected from one particular family of lattice planes in all crystals which are correctly oriented, (c) The form of a powder pattern (asymmetric film mounting)... 

Let us start with a few definitions. A lattice plane of a given 3D BL contains at least three noncollinear lattice points and this plane forms a 2D BL. A family of lattice planes of a 3D BL is a set of parallel equally-spaced lattice planes separated by the minimum distance d between planes and this set contains all the points of the BL. The resolution of a given 3D BL into a family of lattice planes is not unique, but for any family of lattice planes of a direct BL, there are vectors of the reciprocal lattice that are perpendicular to the direct lattice planes. Inversely, for any reciprocal lattice vector G, there is a family of planes of the direct lattice normal to G and separated by a distance d, where 2jt/d is the length of the shortest reciprocal lattice vector parallel to G. A proof of these two assertions can be found in Ashcroft and Mermin [1]. 

In fee and bcc lattices, there are no cubic primitive cells whereas in simple cubic system, the reciprocal lattice is also simple cubic and the Miller indices of a family of lattice planes represent the coordinates of a vector normal to the planes in the usual Cartesian coordinates. As the lattice planes of a fee cubic lattice or a bcc cubic lattice are parallel to those of a sc lattice, it has then been fixed as a rule to define the lattice planes of the fee and bcc cubic lattices as if they were sc lattices with orthogonal primitive vectors of the reciprocal lattice. 

For the determination of the atomic form factor F two experimental methods present themselves. In the one, which has been most widely used, the X-rays fall upon a crystal. Every atom in the crystal lattice functions as a source of scattered waves which interfere with each other. If the wavelength A of the X-rays, the distance a between successive members of a family of lattice planes and the angle of incidence (f) of the X-rays, measured from these lattice planes, obey the relation of Bragg... 

Diffraction by a family of lattice planes that remain undeformed by the presence of the dislocation will not produce an image of the dislocation [128], [161]. 

Each point of the reciprocal lattice represents, as we have said, a family of crystal planes. The different families of planes define the crystal system, which is therefore determined by assigning a triplet of Miller indices to each diffraction spot or in diffraction on a polycrystalline sample, to each diffraction peak. This is referred to as the indexing of the diffraction peaks. The interplanar distance associated with a family of planes with indices (hkl) is a function of these Miller indices and of the cell parameters. We can write ... 

Figure 7.19. Reciprocal lattice mapping. The intensity distribution is measured when the reciprocal lattice node associated with the family of diffracting planes passes through the Ewald sphere...

A plane which passes through three lattice points (and hence through an infinite number of lattice points) is a lattice plane. Planes equivalent by translation form a family of regularly spaced lattice planes. The greater the distance between the planes, the smaller is the area of the primitive two-dimensional cell because all the primitive cells of the lattice have the same volume. Figure 1.10 shows a family of parallel planes numbered consecutively with plane 0 passing... 

The crystal planes pass through an infinite number of equilibrium positions of the atoms. Planes that are parallel to each other form a family of crystal planes (Figure 6.1). For such a family, it holds that all crystal planes are at the same distance from the next plane. Call this distance d. All lattice points will belong to one of the crystal planes of a given family. 

FIGURE 6.1 Two-dimensional crystal as an illustration. Black circles are lattice points. Crystal directions (arrows) define the unit cell (gray). Two members of the (2,1) family of crystal planes are marked out (in two dimension as lines). 

Thus, the vector tf of reciprocal lattice results as perpendicular on crystallographic (hkl) plane and has the 1/ d magnitude properly to the inverse of interplanar distance for family of (hkl) planes. 

Families of equivalent planes, such as planes perpendicular to the lattice vectors, are denoted by curly brackets. For cubic lattices only, the vector perpendicular to a plane has the same indices as the Miller indices of the plane. 

Fig. 7.1. Families of planes passing through lattice points.

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